Dynamic Routing Between Capsules STAT946

Presented by

Yang, Tong(Richard)

Hinton's Critiques on CNN

Four arguments against pooling

• It is a bad fit to the psychology of shape perception: It does not explain why we assign intrinsic coordinate frames to objects and why they have such huge effects.

• It solves the wrong problem: We want equivariance, not invariance. Disentangling rather than discarding.

• It fails to use the underlying linear structure: It does not make use of the natural linear manifold that perfectly handles the largest source of variance in images.

• Pooling is a poor way to do dynamic routing: We need to route each part of the input to the neurons that know how to deal with it. Finding the best routing is equivalent to parsing the image.

Equivariance

• Without the sub-sampling, convolutional neural nets give "place-coded" equivariance for discrete translations.

Two types of equivariance

Place-coded equivariance

If a low-level part moves to a very different position it will be represented by a different capsule.

Rate-coded equivariance

If a part only moves a small distance it will be represented by the same capsule but the pose outputs of the capsule will change.

Higher-level capsules have bigger domains so low-level place-coded equivariance gets converted into high-level rate-coded equivariance.

== Extrapolating shape recognition to very different viewpoints

• Current neural net wisdom:
  • Learn different models for different viewpoints.
  • This requires a lot of training data.
• A much better approach:
  • The manifold of images of the same rigid shape is highly non-linear in the space of pixel intensities.
  • Transform to a space in which the manifold is globally linear

Mathematical Representations

Capsule

\begin{align} s_j = \sum_{i}c_{ij}\hat{u}_{j|i} \end{align}

where

\begin{align} \hat{u}_{j|i} = W_{ij}u_i \end{align}

Two Key Features of Capsule Network

Squashing

\begin{align} v_j = \frac{||s_j||^2}{1+||s_j||^2}\frac{s_j}{||s_j||} \end{align}

Routing By Agreement

\begin{align} c_{ij} = \frac{exp(b_ij)}{\sum_{k}exp(b_ik)} \end{align}